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Capacities in Complex Analysis pp 81–98Cite as

Gamma Capacity

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Part of the book series: Aspects of Mathematics / Aspekte der Mathematik ((ASMA,volume E 14))

Abstract

Definition (the Choquet Integral). Assume that f is a non-negative function and c a capacity. Then \(\int {f{d_C}} \) is defined by

$$\int {f{d_c} = \int\limits_0^\infty {c\left( {\left\{ {x;f\left( x \right) > s} \right\}} \right)ds} } $$

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Notes and references

  • Theorem IX:2 is due to F. Topsøe, On construction of measures. Københavns Universitet, Mat. Inst. Preprint series 1974:27.

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  • Corollary IX:1 is due to G. Choquet, Theory of capacities. Ann. Inst. Fourier 5 (1953-54).

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  • The notation of swarm is closely related to that of “noyau capacitaire regulier” as defined in C. Dellacherie, Ensembles analytiques. Capacités. Mesures de Hausdorff. Springer LNM. 295 (1972).

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  • Example 1 is due to V. Šeinow (see Ronkins book below). Example 2 is due to C.O. Kiselman. Manuscript. Uppsala 1973.

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  • The gamma capacity was introduced in L.I. Ronkin, Introduction to the theory of entire functions of several variables. Amer. Math. Soc. Providence. R.I. 1974, and the modified gamma capacity is in S.Ju. Favorov, On capacity characterizations of sets in ₵n. Charkov 1974 (Russian). The remark by Remmert, used in the proof of Theorem IX:9 is in R. Remmert, Holomorphe und meromorphe Abbildungen komplexer Räume. Math. Ann. 133 (1957).

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  • The set functions γn and Γn has been used in connection with removable singularity sets; cf. U. Cegrell, Removable singularity sets for analytic functions having modulus with bounded Laplace mass. Proc. Amer. Math. Soc. Vol. 88 (1983).

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  • P. Järvi, Removable singularities for Hp-functions. Proc. Amer. Math. Soc. Vol. 86 (1982).

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  • J. Riihentaus, An extension theorem for meromorphic functions of several variables. Ann. Acad. Sc. Fenn. Sér. AI. Vol. 4 (1978/79).

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  • Some of the material of this section has been published in seminaire Pierre Lelong-Henri Skoda (Analyse) 1978/79. Springer LNM 822. 1980.

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Authors and Affiliations

  1. Department of Mathematics, University of Umeå, Sweden

    Prof. Dr. Urban Cegrell

Authors
  1. Prof. Dr. Urban Cegrell
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© 1988 Springer Fachmedien Wiesbaden

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Cegrell, U. (1988). Gamma Capacity. In: Capacities in Complex Analysis. Aspects of Mathematics / Aspekte der Mathematik, vol E 14. Vieweg+Teubner Verlag, Wiesbaden. https://doi.org/10.1007/978-3-663-14203-4_10

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  • DOI: https://doi.org/10.1007/978-3-663-14203-4_10

  • Publisher Name: Vieweg+Teubner Verlag, Wiesbaden

  • Print ISBN: 978-3-528-06335-1

  • Online ISBN: 978-3-663-14203-4

  • eBook Packages: Springer Book Archive

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